lyapunov exponents and spectral analysis of ergodic …kotani).pdf · kotani theory and its...

LYAPUNOV EXPONENTS AND SPECTRAL ANALYSIS OFERGODIC SCHRODINGER OPERATORS: A SURVEY OF

KOTANI THEORY AND ITS APPLICATIONS

DAVID DAMANIK

Dedicated to Barry Simon on the occasion of his 60th birthday.

Abstract. The absolutely continuous spectrum of an ergodic family of one-

dimensional Schrodinger operators is completely determined by the Lyapunov

exponent as shown by Ishii, Kotani and Pastur.Moreover, the part of the theory developed by Kotani gives powerful tools

for proving the absence of absolutely continuous spectrum, the presence of

absolutely continuous spectrum, and even the presence of purely absolutelycontinuous spectrum.

We review these results and their recent applications to a number of prob-

lems: the absence of absolutely continuous spectrum for rough potentials, theabsence of absolutely continuous spectrum for potentials defined by the dou-

bling map on the circle, and the absence of singular spectrum for the subcriticalalmost Mathieu operator.

1. Introduction

Schrodinger operators with ergodic potentials have enjoyed quite some popularityfor several decades now. This is in no small part due to Barry Simon’s contributionsto the field, through research articles on the one hand, but also through surveyarticles and his way of putting his personal stamp on results and conjectures onthe other hand. Ergodic Schrodinger operators continue to be dear to him asseven of the fifteen Schrodinger operator problems he singles out in [60] for furtherinvestigation in the 21st century deal with them. Moreover, the immense activityin the area of ergodic Schrodinger operators is reflected by the fact that the ratio7/15 improves to 3/4 when it comes to the problems from that list that have beensolved so far. One may say that this is due to the uneven distribution of difficultyamong these fifteen problems, but this is balanced by the fact that of the remainingfour ergodic problems at least three are very hard and that further progress shouldbe expected on some of the remaining non-ergodic problems.

In the area of ergodic Schrodinger operators there are several powerful meth-ods (e.g., fractional moment) and analyses (multi-scale) but few theories (Kotani).What appears to be wordplay wants to express the fact that Kotani theory is distin-guished from the other greats by its immensely general scope. It really is a theorythat applies to the class of all ergodic operators and it is central in many ways. In

Date: May 18, 2006.

2000 Mathematics Subject Classification. Primary 82B44, 47B80; Secondary 47B36, 81Q10.Key words and phrases. Ergodic Schrodinger Operators, Lyapunov Exponents.The author was supported in part by NSF grant DMS–0500910.

1

2 DAVID DAMANIK

addition, Kotani theory has played a crucial role in the solution of two of the fourrecently solved 21st century problems.

Our goal here is to present the core parts of Kotani theory with more or lesscomplete proofs and to discuss several recent applications of the theory to a numberof concrete classes of models for which, whenever possible, we at least outline themain ideas that go into the proofs of the results we mention.

Suppose (Ω, µ) is a probability measure space, T : Ω → Ω is an invertible ergodictransformation, and f : Ω → R is bounded and measurable. We define potentials,

Vω(n) = f(Tnω), ω ∈ Ω, n ∈ Z,

and the corresponding discrete Schrodinger operators in `2(Z),

(1) [Hωψ](n) = ψ(n+ 1) + ψ(n− 1) + Vω(n)ψ(n).

We will call Hωω∈Ω an ergodic family of Schrodinger operators.

Examples. (a) Quasi-periodic potentials: Ω = Td = Rd/Zd, µ is the normalizedLebesgue measure on Td, and Tω = ω + α is some ergodic shift (i.e., 1, α1, . . . , αd

are rationally independent).1

(b) Potentials defined by the skew shift: Ω = T2, µ is the normalized Lebesguemeasure on T2, and T (ω1, ω2) = (ω1 + ω2, ω2 + α) for some irrational α.(c) Potentials defined by the doubling map: Ω = T, µ is the normalized Lebesguemeasure on T, and Tω = 2ω.2

(d) Potentials defined by the left shift: Ω = IZ, where I is a compact subset of R,µ = PZ, where P is a Borel probability measure on I, and [Tω](n) = ω(n+ 1).

The following pair of results, proven in [50, 56], shows that for ergodic families ofSchrodinger operators, the spectrum and the spectral type are deterministic in thesense that they are constant µ-almost surely.

Theorem 1 (Pastur 1980). There exists a set Σ ⊂ R such that for µ-almost everyω, σ(Hω) = Σ.

Theorem 2 (Kunz-Souillard 1980). There are sets Σac, Σsc, Σpp ⊂ R such thatfor µ-almost every ω, σ•(Hω) = Σ•, • ∈ ac, sc,pp.

Thus, in the spectral analysis of a given ergodic family of Schrodinger operators,a fundamental problem is the identification of the sets Σ, Σac, Σsc, and Σpp.

The almost sure spectrum, Σ, is completely described by the integrated densityof states as shown by Avron and Simon [9]. Denote the restriction ofHω to [0, N−1]with Dirichlet boundary conditions byH(N)

ω . For ω ∈ Ω andN ≥ 1, define measuresdkω,N by placing uniformly distributed point masses at the eigenvalues E(N)

ω (1) <

1What we call quasi-periodic here is more general than the notion of quasi-periodicity as defined

in [7], for example, where a quasi-periodic potential is almost periodic with a finitely generatedfrequency module. In particular, a quasi-periodic potential as defined here is not necessarilyalmost periodic, that is, the translates of a given quasi-periodic potential are not necessarily

precompact in `∞(Z). We allow discontinuous sampling functions f here because we want toinclude Fibonacci-type potentials.

2Strictly speaking, this example does not fall within our general framework as T is not invert-ible, but potentials of this kind have been studied in several works and it is possible to tweak the

model a little to fit it in the framework above.

A SURVEY OF KOTANI THEORY 3

· · · < E(N)ω (N) of H(N)

ω , that is,∫f(E) dkω,N (E) =

1N

N∑n=1

f(E(N)ω (n)).

Then, it can be shown that for µ-almost every ω ∈ Ω, the measures dkω,N convergeweakly to a non-random measure dk, called the density of states measure, as N →∞. The function k defined by

k(E) =∫χ(−∞,E](E′) dk(E′)

is called the integrated density of states. It is not hard to show that

(2)∫f(E) dk(E) = E (〈δ0, f(Hω)δ0〉)

for bounded measurable f . Here, E(·) denotes integration with respect to themeasure µ, that is, E(g) =

∫g(ω) dµ(ω) Thus, the density of states measure is

given by an average of the spectral measures associated with Hω and δ0. TheT -invariance of µ then implies the following result:

Theorem 3 (Avron-Simon 1983). The almost sure spectrum is given by the pointsof increase of k, that is, Σ = supp(dk).

There is a similarly general description of the set Σac in terms of the Lyapunovexponent. Let E ∈ C and

(3) AE(ω) =(E − f(ω) −1

1 0

).

Define the Lyapunov exponent γ(E) by

γ(E) = limn→∞

1n

E(log ‖AE

n (ω)‖),

where AEn (ω) = AE(Tn−1ω) · · ·AE(ω).

The integrated density of states and the Lyapunov exponent are related by theThouless formula (see, e.g., [23, Theorem 9.20]), which reads

(4) γ(E) =∫

log |E − E′| dk(E′).

The significance of the transfer matrices AEn (ω) is that a sequence u solves the

difference equation

(5) u(n+ 1) + u(n− 1) + Vω(n)u(n) = Eu(n)

if and only if (un

un−1

)= AE

n (ω)(u0

u−1

)for every n. Since detAE(ω) = 1, we always have γ(E) ≥ 0. Let us define

Z = E : γ(E) = 0

By general principles, Z ⊆ Σ.For a subset S of R, the essential closure of S is given by

Sess

= E ∈ R : Leb(S ∩ (E − ε, E + ε)) > 0 for every ε > 0

4 DAVID DAMANIK

Then, the following theorem combines results from [37, 43, 56].3

Theorem 4 (Ishii 1973, Pastur 1980, Kotani 1984). The almost sure absolutelycontinuous spectrum is given by the essential closure of the set of energies for whichthe Lyapunov exponent vanishes, that is, Σac = Zess

.

While there is an analog of Theorem 3 for higher-dimensional ergodic Schrodingeroperators, Theorem 4 is, by its very nature, a strictly one-dimensional result. It isone of the great challenges for researchers in the area of ergodic Schrodinger opera-tors to develop effective tools for the study of the absolutely continuous spectrum inhigher dimensions. That said, Theorem 4 holds in virtually all one-dimensional andquasi-one-dimensional situations: see Kotani [43] for continuous one-dimensionalSchrodinger operators (see also Kirsch [41] for a useful extension), Minami [54] forgeneralized Sturm-Liouville operators, Kotani-Simon [49] for discrete and contin-uous Schrodinger operators with matrix-valued potentials, and Geronimo [31] andGeronimo-Teplyaev [32] for orthogonal polynomials on the unit circle.

2. The Description of the AC Spectrum

In this section we discuss the main ideas that go into the proof of Theorem 4.The proof naturally breaks up into the proof of two inclusions.

The inclusion “⊆” was proved by Ishii and Pastur. The other inclusion wasproved by Kotani and is a much deeper result. In fact, the proof of the Ishii-Pastur half of the result we give below is based on more modern techniques andshows that this half is really an immediate consequence of the general theory ofone-dimensional Schrodinger operators.

There are at least three different proofs of the Ishii-Pastur half of Theorem 4 inthe literature. One of them uses the existence of generalized eigenfunctions; com-pare Cycon et al. [23]. The second one, due to Deift and Simon [30], is close in spiritto, and uses techniques from, Kotani’s proof of the other half of the result. Finally,there are two somewhat related proofs given by Buschmann [18] and Last and Si-mon [51], which are both either directly or indirectly based on a result of Gilbertand Pearson that describes a support of the singular spectrum of a Schrodingeroperator with fixed (non-random) potential in terms of subordinate solutions. Wewill follow the argument from Buschmann’s paper below.

We first recall a central result from Gilbert and Pearson’s subordinacy theory[33, 34]. Consider the discrete Schrodinger operator H in `2(Z) with potential Vand solutions of the difference equation

(6) u(n+ 1) + u(n− 1) + V (n)u(n) = Eu(n).

A non-zero solution u of (6) is called subordinate at ±∞ if for every linearly inde-pendent solution u of (6), we have∑N−1

n=0 |u(±n)|2∑N−1n=0 |u(±n)|2

→ 0 as N →∞.

Let

S = E ∈ R : (6) has solutions u+ and u− such that u± is subordinate at ±∞.

3To be more precise, the discrete version of the Kotani half of this result here can be found inthe paper [59] by Simon and the work of Ishii and Pastur was preceded by closely related work

by Casher and Lebowitz [19].


Then, S has zero weight with respect to the absolutely continuous part of anyspectral measure, that is,

(7) P(ac)(S) = 0.

Proof of the Ishii-Pastur half of Theorem 4. Note that when γ(E) > 0, Oseledec’Theorem [55] says that for almost every ω, there are solutions u+(E,ω) andu−(E,ω) of (5) such that u±(E,ω) is exponentially decaying, and hence subor-dinate, at ±∞. Applying Fubini’s theorem, we see that for µ-almost every ω, theset of E ∈ R \ Z for which the property just described fails, has zero Lebesguemeasure. In other words, for these ω’s, R \ Z ⊆ Sω up to a set of zero Lebesguemeasure. Since sets of zero Lebesgue measure have zero weight with respect to theabsolutely continuous part of any spectral measure, we obtain from (7) that forµ-almost every ω,

P(ac)ω (R \ Z) = 0.

This shows that for µ-almost every ω, σac(Hω) ⊆ Zess.

Let us now turn to the Kotani half of Theorem 4. Kotani worked in the contin-uum setting. Carrying his results over to the discrete case is not entirely straight-forward and it was worked out by Simon [59] whose proof we give below. Givenz ∈ C+ = z ∈ C : =z > 0 and ω ∈ Ω, there are (up to a multiplicative constant)unique solutions u±(n, ω) of

(8) u(n+ 1) + u(n− 1) + Vω(n)u(n) = zu(n)

such that u± is square-summable at ±∞. (Take u±(n, ω) = 〈δn, (Hω−z)−1δ1〉 near±∞ to show existence; uniqueness follows from constancy of the Wronskian.) Notethat u±(0, ω) 6= 0 for otherwise z would be a non-real eigenvalue of a self-adjointhalf-line operator. Thus, we can define

(9) m±(z, ω) = −u±(±1, ω)u±(0, ω)

.

Clearly,

(10) m±(z, Tnω) = −u±(n± 1, ω)u±(n, ω)

.

By Oseledec’ Theorem, we have for µ-almost every ω,

limn→∞

1n

log∣∣∣∣u±(n, ω)u±(0, ω)

∣∣∣∣ = −γ(z).

By (10),

log∣∣∣∣u±(n, ω)u±(0, ω)

∣∣∣∣ = n−1∑m=0

log |m±(z, T±mω)|

and hence Birkhoff’s ergodic theorem implies

(11) E(log |m±(z, ω)|) = −γ(z).

Proposition 2.1. We have that

E(

log(

1 +=z

=m±(z, ω)

))= 2γ(z).

6 DAVID DAMANIK

Proof. By the difference equation (8) that u± obeys,

(12) m±(z, Tnω) = Vω(n)− z − [m±(z, Tn∓1ω)]−1.

Taking imaginary parts,

=m±(z, ω) = −=z −=([m±(z, T∓1ω)]−1

).

Dividing by =m+(z, ω),

1 = − =z=m±(z, ω)

−=([m±(z, T∓1ω)]−1

)=m±(z, ω)

.

Taking the logarithm,

log(

1 +=z

=m±(z, ω)

)= log

(−=

([m±(z, T∓1ω)]−1

))− log (=m±(z, ω))

= log(=m±(z, T∓1ω)|m±(z, T∓1ω)|2

)− log (=m±(z, ω))

Taking expectations and using invariance,

E(

log(

1 +=z

=m±(z, ω)

))= E

(log(=m±(z, T∓1ω)|m±(z, T∓1ω)|2

)− log (=m±(z, ω))

)= −2 E (log |m±(z, ω)|)= 2γ(z),

where we used (11) in the last step.

Denote

b(z, ω) = m+(z, ω) +m−(z, ω) + z − Vω(0).

Proposition 2.2. We have

E(=(

1b(z, ω)

))= − ∂γ(z)

∂(=z).

Proof. It follows from (12) that

(13)u−(1, ω)u−(0, ω)

= m−(z, ω) + z − Vω(0).

It is not hard to check that for n ≤ m,

(14) Gω(n,m; z) := 〈δn, (Hω − z)−1δm〉 =u−(n, ω)u+(m,ω)

u+(1, ω)u−(0, ω)− u−(1, ω)u+(0, ω).

From (9), (13), (14), we get

(15) −Gω(0, 0; z)−1 = m+(z, ω) +m−(z, ω) + z − Vω(0) = b(z, ω).

The definition of Gω(n,m; z) gives

(16) E(Gω(0, 0; z)) =∫

1E′ − z

dk(E′).


Thus,

E(=(

1b(z, ω)

))= −=E (Gω(0, 0; z))

= −=∫

1E′ − z

dk(E′)

= − ∂

∂(=z)

∫log |z − E′| dk(E′)

= − ∂γ(z)∂(=z)

,

where we used (15), (16), and the Thouless formula (4).

Denote

n±(z, ω) = =m±(z, ω) + 12=z.

Proposition 2.3. We have that

(17) E(

1n±(z, ω)

)≤ 2γ(z)

=z

and

(18) E

[

1n+

+ 1n−

]·[(n+ − n−)2 + (<b)2

]|b|2

≤ 4[γ(z)=z

− ∂γ(z)∂(=z)

].

Proof. For x ≥ 0, consider the function A(x) = log(1+x)− x1+ x

2. Clearly, A(0) = 0

and A′(x) = 11+x −

1

1+x+ x24

≥ 0. Therefore,

(19) log(1 + x) ≥ x

1 + x2

for all x ≥ 0.

Thus,

E(

1n±(z, ω)

)= E

(1

=m±(z, ω) + 12=z

)

=1=z

E

=z=m±(z,ω)

1 +=z

=m±(z,ω)

2

≤ 1=z

E(

log(

1 +=z

=m±(z, ω)

))=

2γ(z)=z

,

which is (17). We used (19) in the third step and Proposition 2.1 in the last step.

8 DAVID DAMANIK

Notice that n+(z, ω)+n−(z, ω) = =b(z, ω). Thus the integrand on the left-handside of (18) is equal to[

1n+

+1n−

]·[(n+ + n−)2 − 4n+n− + (<b)2

]|b|2

=

[1n+

+1n−

]·[|b|2 − 4n+n−

]|b|2

=1n+

+1n−

− 4n+ + n−|b|2

=1n+

+1n−

+ 4=(

1b

).

The bound (18) now follows from (17) and Proposition 2.2.

Proof of the Kotani half of Theorem 4. The Thouless formula (4) says that

γ(z) =∫

log |z − E′| dk(E′) = <∫

log(z − E′) dk(E′)

and hence −γ(z) is the real part of a function whose derivative is a Borel transform(namely, of the measure dk). By general properties of the Borel transform, it followsthat the limit γ′(E+ i0) exists for Lebesgue almost every E ∈ R and, in particular,for almost every E ∈ Z. For these E, we have that

(20) limε↓0

γ(E + iε)ε

= limε↓0

γ(E + iε)− γ(E)ε− 0

= limε↓0

∂γ

∂(=z)(E + iε),

and in particular, the limit is finite. Thus, by (17),

lim supε↓0

E(

1=m±(E + iε, ω)

)<∞

for almost every E ∈ Z. Since m± are Borel transforms as well (of the spectralmeasures associated with half-line restrictions of Hω), we also have that, for everyω ∈ Ω, m±(E + i0, ω) exists for Lebesgue almost every E ∈ R, and hence, foralmost every E, m±(E + i0, ω) exists for almost every ω. Combining the last twoobservations with Fatou’s lemma, we find that

(21) E(

1=m±(E + i0, ω)

)<∞

for almost every E in Z. So, for almost every ω ∈ Ω and E ∈ Z, =m±(E+i0, ω) > 0.On the other hand, m+(E + iε, ω) +m−(E + iε, ω) +E + iε− Vω(0) has a finite

limit for almost every ω ∈ Ω and E ∈ Z.Hence, (15) shows that 0 < =Gω(0, 0;E + i0) < ∞ for almost every ω ∈ Ω and

E ∈ Z, which implies the result.

Denote the measure associated with the Herglotz function Gω(0, 0; z) by νω, thatis,

Gω(0, 0; z) =∫dνω(E)E − z

.

The results above imply the following for µ-almost every ω:

ν(ac)ω (E) = 0 for Lebesgue almost every E ∈ R \ Z,

ν(ac)ω (E) > 0 for Lebesgue almost every E ∈ Z.


Here, ν(ac)ω (E) denotes the density of the absolutely continuous part of νω. Write

k(ac)(E) for the density of the absolutely continuous part of the density of statesmeasure.

There is a direct relation between these densities [47]:

Theorem 5 (Kotani 1997). For almost every E ∈ Z,

(22) k(ac)(E) = E(ν(ac)

ω (E)).

Proof. The inequality “≥” in (22) follows from (2) (i.e., the density of states mea-sure is the average of the measures νω) and the fact that the average of absolutelycontinuous measures is absolutely continuous.

To prove the opposite inequality, we first note that for almost every E ∈ Z, (16),(20), and Cauchy-Riemann imply

(23) k(ac)(E) =1π

limε↓0

γ(E + iε)ε

.

Because of (20), (21) and Fatou’s lemma, (18) implies that for almost every pair(E,ω) ∈ Z × Ω,

(24) =m+(E + i0, ω) = =m−(E + i0, ω)

and

(25) <m+(E + i0, ω) + <m−(E + i0, ω) + E − Vω(0) = 0.

Thus, for almost every (E,ω) ∈ Z × Ω,

ν(ac)ω (E) =

1π=Gω(0, 0;E + i0)(26)

=1π= −1m+(E + i0, ω) +m−(E + i0, ω) + E − Vω(0)

=1π= −1

2i=m+(E + i0, ω)

=12π

1=m+(E + i0, ω)

Let Pε be the Poisson kernel for the upper half-plane, that is,

Pε(E) =1π

ε

E2 + ε2.

Write

Cε(E) =∫ZPε(E − E′) dE′

and

Pε(E,E′) = Pε(E − E′)Cε(E)−1.

10 DAVID DAMANIK

Then, by (26) and Jensen’s inequality, we obtain for almost every (E,ω) ∈ Z × Ω,∫Rν(ac)

ω (E′)Pε(E − E′) dE′ ≥∫Zν(ac)

ω (E′)Pε(E − E′) dE′

=∫Z

(12π

1=m+(E′ + i0, ω)

)Pε(E − E′) dE′

= Cε(E)∫Z

(12π

1=m+(E′ + i0, ω)

)Pε(E,E′) dE′

≥ Cε(E)

(∫Z

(12π

1=m+(E′ + i0, ω)

)−1

Pε(E,E′) dE′)−1

≥ Cε(E)2(∫

R

(12π

1=m+(E′ + i0, ω)

)−1

Pε(E − E′) dE′)−1

=Cε(E)2

2π1

=m+(E + iε, ω)

Thus, for almost every E ∈ Z,∫R

E(ν(ac)

ω (E′))Pε(E − E′) dE′ ≥ Cε(E)2

2πE(

1=m+(E + iε, ω)

),

and hence

(27) E(ν(ac)

ω (E))≥ 1

2πlim sup

ε↓0E(

1=m+(E + iε, ω)

)since Cε(E) < 1 and Cε(E) → 1 as ε ↓ 0.

Using (23), Proposition 2.1, the inequality log(1 + x) ≤ x for x ≥ 0, and then(27), we find that

k(ac)(E) =1π

limε↓0

γ(E + iε)ε

=1π

limε↓0

12ε

E(

log(

1 +ε

=m±(E + iε, ω)

))≤ 1

2πlim sup

ε↓0E(

1=m±(E + iε, ω)

)≤ E

(ν(ac)

ω (E)),

concluding the proof of “≤” in (22).

Corollary 1. The spectrum is almost surely purely absolutely continuous if andonly if the integrated density of states is absolutely continuous and the Lyapunovexponent vanishes almost everywhere with respect to the density of states measure.

There is a different approach to purely absolutely continuous spectrum as pointedout by Yoram Last (unpublished):4 Using a result of Deift and Simon [30, The-orem 7.1], one can show that there is a set R ⊆ Z such that Leb(Z \ R) = 0and R has zero singular spectral measure for every ω ∈ Ω due to the absence ofsubordinate solutions.

4The author is grateful to Barry Simon for bringing this to his attention.


The spectrum naturally breaks up into the two components Z and Σ \ Z. It isknown that either set can support singular continuous spectrum as demonstratedby the almost Mathieu operator at critical and super-critical coupling. However,as we have seen, Σ \ Z does not support any absolutely continuous part of thespectral measures. Trivially, Anderson localization is impossible in Z. However, itis tempting to expect even more:

Problem 1. Prove or disprove that for all ergodic families, the operators Hω haveno eigenvalues in Z.

If the answer is affirmative, this will in particular imply the absence of eigenvaluesin a number of special cases, such as the operators considered in Section 6 and thecritical almost Mathieu operator.5

3. The Induced Measure and its Topological Support

Fix a compact subset R of R. Endow RZ with product topology, which makesit a compact metric space. If V ∈ RZ, we define the functions m± by

m±(z) = ∓u±(1)u±(0)

,

where u± solves

(28) u(n+ 1) + u(n− 1) + V (n)u(n) = zu(n)

and is `2 at ±∞.Denote Z+ = 1, 2, 3, . . . and Z− = 0,−1,−2, . . .. It is well known that the

maps M± : V± = V |Z± 7→ m± are one-one and continuous with respect to uniformconvergence on compacta on the m-function side. The m-functions m± are Herglotzfunctions, they have boundary values almost everywhere on the real axis, and theyare completely determined by their boundary values on any set of positive Lebesguemeasure.

We will be interested in those V for which the functions m+,m− obey identitieslike (24) and (25), that is,

(29) m−(E + i0) = −m+(E + i0)

for a rich set of energies. Thus, for a set Z ⊆ R, we let

D(Z) = V ∈ RZ : m± associated with V obey (29) for a.e. E ∈ Z.

On RZ, define the shift transformation [S(V )](n) = V (n+ 1).

Lemma 3.1. Suppose that Z ⊂ R has positive Lebesgue measure. Then:(a) D(Z) is S-invariant and closed in RZ.(b) For V ∈ D(Z), denote the restrictions to Z± by V±. Then V− determines V+

uniquely among elements of D(Z) and vice versa.(c) If there exist V (m), V ∈ D(Z) such that V (m)

− → V− pointwise, then V (m)+ → V+

pointwise.

5For this particular model, this would provide an alternative to the proof based on self-dualityand zero-measure spectrum.

12 DAVID DAMANIK

Proof. (a) If u1, u2 denote the solutions of (28) that obey u1(0) = u2(1) = 1 andu1(1) = u2(0) = 0, then we can write (note that we may normalize u± by u±(0) = 1)

u±(n) = u1(n)∓m±(z)u2(n).

Let us denote the m-functions associated with S(V ) by m±. Clearly,

m±(z) = ∓u1(2)∓m±(z)u2(2)u1(1)∓m±(z)u2(1)

.

Since the uj(m) are polynomials in z with real coefficients, this shows that

m−(E + i0) = −m+(E + i0)

for almost every E ∈ Z and hence D(Z) is S-invariant. It follows from the conti-nuity of the maps M± that D(Z) is closed. (For a proof that the identity betweenthe boundary values of the associated m-functions is preserved after taking limits,see [44, Lemma 5] and [45, Lemma 7.4].)(b) V− determines m− and then (29) determines the boundary values of m+ on aset of positive Lebesgue measure. By general properties of Herglotz functions, thisdetermines m+ (and hence V+) completely. By the same argument, V+ determinesV−.(c) By compactness, there is a subsequence of V (m) that converges pointwise,that is, there is V such that V (mk) → V as k → ∞. By part (a), V ∈ D(Z). Byassumption, V− = V−. Thus, by part (b), V+ = V+, and hence V = V . Con-sequently, V (mk)

+ → V+ pointwise. In fact, we claim that V (m)+ → V+ pointwise.

Otherwise, we could reverse the argument (i.e., go from right to left) and show thatV

(mk)− 6→ V− for some other subsequence.

We will now derive two important consequences of Lemma 3.1: the absence ofabsolutely continuous spectrum for topologically non-deterministic families and thesupport theorem. To do so, we will consider the push-forward ν of µ on the sequencespace and its topological support.

More precisely, given an ergodic dynamical system (Ω, µ, T ) and a measurablebounded sampling function f : Ω → R defining potentials Vω(n) = f(Tnω) asbefore, we associate the following dynamical system (RZ, ν, S): R is a compactset that contains the range of f , ν is the Borel measure on RZ induced by µ viaΦ(ω) = Vω (i.e., ν(A) = µ(Φ−1(A))), and S is the shift transformation on RZ

introduced above. Recall that the topological support of ν, supp ν, is given by theintersection of all compact sets B ⊆ RZ with ν(B) = 1. Clearly, supp ν is closedand S-invariant.

Theorem 6 (Kotani 19896). Let (Ω, µ, T, f) and (RZ, dν, S) be as just described.Assume that the set

Z = E ∈ R : γ(E) = 0.has positive Lebesgue measure. Then,(a) Each V ∈ supp ν is determined completely by V− (resp., V+).(b) If we let

(supp ν)± = V± : V ∈ supp ν,

6The result appears explicitly in the 1989 paper [46]. The main ingredients of the proof,however, were found earlier [44, 45].


then the mappings

(30) E± : (supp ν)± 3 V± 7→ V∓ ∈ (supp ν)∓are continuous with respect to pointwise convergence.

Proof. (a) By our earlier results, we know that D(Z) is compact and has full ν-measure. Thus, supp ν ⊆ D(Z) and the assertion follows from Lemma 3.1.(b).(b) This follows from Lemma 3.1.(c).

We say that (Ω, µ, T, f) is topologically deterministic if there exist continuousmappings E± : (supp ν)± → (supp ν)∓ that are formal inverses of one another andobey V #

− ∈ supp ν for every V− ∈ (supp ν)−, where

V #− (n) =

V−(n) n ≤ 0,E−(V−)(n) n ≥ 1.

This also implies V #+ ∈ supp ν for every V+ ∈ (supp ν)+, where

V #+ (n) =

V+(n) n ≥ 1,E+(V+)(n) n ≤ 0.

Otherwise, (Ω, µ, T, f) is topologically non-deterministic.

Corollary 2. If (Ω, µ, T, f) is topologically non-deterministic, then Σac = ∅.

Our next application of Lemma 3.1 is the so-called support theorem; compare[44]. For a Borel measure ν on RZ, let Σac(ν) ⊆ R denote the almost sure absolutelycontinuous spectrum, that is, σac(∆+V ) = Σac(ν) for ν almost every V . If ν comesfrom (Ω, µ, T, f), then Σac(ν) coincides with the set Σac introduced earlier. Thesupport theorem says that Σac(ν) is monotonically decreasing in the support of ν.

Theorem 7 (Kotani 1985). For every V ∈ supp ν, we have σac(∆ + V ) ⊇ Σac(ν).In particular, supp ν1 ⊆ supp ν2 implies that Σac(ν1) ⊇ Σac(ν2).

Proof. We know that

supp ν ⊆ D(Z) = V ∈ RZ : m± associated with V obey (29) for a.e. E ∈ Z.Bearing in mind the Riccati equation (13), a calculation like the one in (26) thereforeshows that for every V ∈ supp ν, the Green function associated with the operator∆ + V obeys =G(0, 0;E + i0) > 0 for almost every E ∈ Z. This implies Zess ⊆σac(∆ + V ) and hence the result by Theorem 4.

A different proof may be found in Last-Simon [51, Sect. 6]. Here is a typicalapplication of the support theorem:

Corollary 3. Let Perν be the set of V ∈ supp ν that are periodic, that is, SpV = Vfor some p ∈ Z+. Then,

Σac(ν) ⊆⋂

V ∈Perν

σ(∆ + V ).

If there are sufficiently many gaps in the spectra of these periodic operators, onecan show in this way that Σac(ν) is empty.

Corollaries 2 and 3 have been used in a variety of scenarios to prove the absenceof absolutely continuous spectrum. In fact, while the Kotani half of Theorem 4concerns the presence of absolutely continuous spectrum on Z, it could be argued

14 DAVID DAMANIK

that the criteria for the absence of absolutely continuous spectrum that are by-products of the theory have been applied more often. This is to a certain extentdue to the fact that the majority of the ergodic families of Schrodinger operators areexpected to have no absolutely continuous spectrum. The following “conjecture” istempting because it is supported by a plethora of results, both on the positive sideand on the negative side. It has been verbally suggested by Yoram Last and it hasappeared explicitly in print in several places, including [39, 48].

Problem 2. Show that Leb (Z) > 0 implies almost periodicity, that is, the closurein `∞(Z) of the set of translates of Vω is compact.

Namely, the presence of (purely) absolutely continuous spectrum is known forall periodic potentials, many limit-periodic potentials,7 and some quasi-periodicpotentials (that are all uniformly almost periodic). On the other hand, the absenceof absolutely continuous spectrum is known for large classes of non-almost periodicergodic potentials. We will see some instances of the latter statement below. Nev-ertheless, proving this conjecture is presumably very hard and it would already beinteresting to find further specific results that support the conjecture. For exam-ple, it is an open (and seemingly hard) problem to prove the absence of absolutelycontinuous spectrum for potentials defined by the skew shift.

We close this section with a problem concerning strips. As was mentioned at theend of the Introduction, Kotani and Simon developed the analog of Kotani theoryfor discrete and continuous Schrodinger operators with matrix-valued potentials intheir 1988 paper [49]. This framework includes in particular discrete Schrodingeroperators on strips. That is, operators of the form ∆ + Vω on `2(Z × 1, . . . , L),where ∆ is again given by the summation over nearest neighbors. Transfer ma-trices are now 2L × 2L and, modulo symmetry, there are L Lyapunov exponents,γL(E) ≥ γL−1(E) ≥ · · · ≥ γ1(E) ≥ 0. For the general matrix-valued situation,they proved that the largest Lyapunov exponent is positive for almost every energyif the potential is non-deterministic. This result cannot be improved in this generalsetting. However, it is reasonable to expect that for strips, the following resultshould hold.

Problem 3. Prove that for non-deterministic Schrodinger operators on a strip, allLyapunov exponents are non-zero for Lebesgue almost all energies.

4. Potentials Generated by the Doubling Map

In this section we discuss potentials defined over the doubling map, that is,Example (c) from the introduction. The underlying dynamical system is stronglymixing and one would hope that the spectral theory of the associated operators isakin to that of the Anderson model, where the potentials are generated by inde-pendent, identically distributed random variables. Alas, by dropping independenceone loses the availability of most tools that have proven useful in the study of theAnderson model.

While localization is expected for Schrodinger operators with potentials overthe doubling map, it has not been shown to hold in reasonable generality. Thereare only two localization results in the literature, and each of them is to some

7A sequence V is limit-periodic if there are periodic sequences V (m) such that ‖V −V (m)‖∞ →0 as m → ∞. See, for example, Avron-Simon [7], Chulaevsky [20], Chulaevsky-Molchanov [21],

and Pastur-Tkachenko [58].


extent unsatisfactory. The first result was found by Bourgain and Schlag [17],who proved localization at small coupling and away from small intervals about theenergies ±2 and 0. Both assumptions seem unnatural. The other result is due toDamanik and Killip [27], who proved localization for essentially all f but only forLebesgue almost every boundary condition at the origin (recall that we are dealingwith operators in `2(Z+)). A result holding for fixed boundary condition would ofcourse be more desirable. To this end, Damanik and Killip were at least able toshow the absence of absolutely continuous spectrum for fixed boundary conditionin complete generality. These results are indeed immediate consequences of Kotanitheory and spectral averaging and we give the short proofs below for the reader’sconvenience.

The first step in a localization proof for a one-dimensional Schrodinger operatoris typically a proof of positive Lyapunov exponents for many energies. For theAnderson model, this can be done for all energies using Furstenberg’s theorem orfor Lebesgue almost all energies using Kotani theory. At small coupling there isalso a perturbative approach due to Pastur and Figotin [57]. The extension of theapproach based on Furstenberg’s theorem to potentials generated by the doublingmap is not obvious; see, however, [4]. The perturbative approach extends quitenicely as shown by Chulaevsky and Spencer [22]. Their results form the basis forthe proof of the partial localization result in [17]. Finally, the approach based onKotani theory also extends as we will now explain.

Theorem 8 (Damanik-Killip 2005). Suppose that f ∈ L∞(T) is non-constantand Vω(n) = f(2nω) for n ≥ 1. Then, the Lyapunov exponent γ(E) is positivefor Lebesgue almost every E ∈ R and the absolutely continuous spectrum of theoperator Hω in `2(Z+) is empty for Lebesgue almost every ω ∈ T.

Proof. Since the proof of this result is so short, we reproduce it here in its entirety.The first step is to conjugate the doubling map T to a symbolic shift via the binaryexpansion. Let Ω+ = 0, 1Z+ and define D : Ω+ → T by D(ω) =

∑∞n=1 ωn2−n.

The shift transformation, S : Ω+ → Ω+, is given by (Sω)n = ωn+1. Clearly,D S = T D.

Next we introduce a family of whole-line operators as follows. Let Ω = 0, 1Z

and define, for ω ∈ Ω, the operator

[Hωφ](n) = φ(n+ 1) + φ(n− 1) + Vω(n)φ(n)

in `2(Z), where

Vω(n) = f [D(ωn, ωn+1, ωn+2, . . .)].

The family Hωω∈Ω is non-deterministic since Vω restricted to Z+ only dependson ωnn≥1 and hence, by non-constancy of f , we cannot determine the values ofVω(n) for n ≤ 0 uniquely from the knowledge of Vω(n) for n ≥ 1. It follows fromCorollary 2 that the Lyapunov exponent for Hωω∈Ω is almost everywhere positiveand σac(Hω) is empty for almost every ω ∈ Ω with respect to the ( 1

2 ,12 )-Bernoulli

measure on Ω.Finally, let us consider the restrictions of Hω to `2(Z+), that is, let H+

ω =E∗HωE, where E : `2(Z+) → `2(Z) is the natural embedding. Observe that H+

ω =Hω, where ω = D(ω1, ω2, ω2, . . .). This immediately implies the statement onthe positivity of the Lyapunov exponent for the family Hωω∈T. As finite-rank

16 DAVID DAMANIK

perturbations preserve absolutely continuous spectrum, σac(H+ω ) ⊆ σac(Hω) for

every ω ∈ Ω. This proves that σac(H+ω ) = ∅ for almost every ω ∈ Ω.

Given φ ∈ (−π/2, π/2), let H(φ)ω denote the operator which acts on `2(Z+) as

in (1), but with ψ(0) given by cos(φ)ψ(0) + sin(φ)ψ(1) = 0. Thus, the originaloperator (with a Dirichlet boundary condition) corresponds to φ = 0. Theorem 8implies the following result for this family of operators:

Corollary 4. Suppose that f is measurable, bounded, and non-constant. Then, foralmost every φ ∈ (−π/2, π/2) and almost every ω ∈ T, the operator H(φ)

ω in `2(Z+)with potential Vω(n) = f(2nω) has pure point spectrum and all eigenfunctions decayexponentially at infinity.

Proof. This is standard and follows quickly from spectral averaging; see, for exam-ple, [57, Theorem 13.4] or [61, Section 12.3].

For a localization proof without the need for spectral averaging, it will be neces-sary to prove the positivity of the Lyapunov exponent for a larger set of energies.Sufficient, for example, is positivity away from a discrete set of exceptional energies.For moderately small coupling, such a result will be contained in [4]. The problemfor other values of the coupling constant is still open.

Problem 4. Find a class of functions f ∈ L∞(T) such that for every λ 6= 0, theLyapunov exponent associated with the potentials Vω(n) = λf(2nω) is positive awayfrom a (λ-dependent) discrete set of energies.

In connection with this problem, important obstructions have been found byBochi [10] and Bochi and Viana [11]. Namely, positivity of γ(E) away from a dis-crete set will fail generically in C(T) (this result holds for rather general underlyingdynamics) and hence the Holder continuity assumptions made in [4] and [22] arenatural.

In some sense a large value of λ alone should ensure the positivity of the Lya-punov exponent. This is indeed the basis of several results for quasi-periodic poten-tials or potentials generated by the skew-shift. For hyperbolic base transformationssuch as the doubling map, however, there is a competition between two differentkinds of hyperbolic behavior that presents problems that have not been solvedyet. Moreover, in the large coupling regime, it would be especially interesting toprove uniform (in energy) lower bounds on the Lyapunov exponents along with thenatural log λ asymptotics.

Problem 5. Find a class of functions f ∈ L∞(T) such that for every λ ≥ λ0(g),the Lyapunov exponent associated with the potentials Vω(n) = λf(2nω) obeysinfE γ(E) ≥ c log λ for some suitable positive constant c.

In this context it should be noted that Herman’s subharmonicity proof fortrigonometric polynomials over ergodic shifts on the torus, [36], works in the caseof the doubling map.8 It seems much less clear, however, how to carry over theSorets-Spencer proof for real-analytic f , [62], from the case of irrational rotationsof the circle to the case of the doubling map, let alone the proof of Bourgain forreal-analytic functions over ergodic shifts on higher-dimensional tori [14].

8The author is grateful to Kristian Bjerklov for pointing this out.


For SL(2,R) cocycles over the doubling map that are not of Schrodinger form(i.e., with a general SL(2,R) matrix replacing (3)), Young has developed a methodfor proving positive Lyapunov exponents “at large coupling” that works in the C1

category [64]. Her method does not immediately apply to Schrodinger cocycles, butit would be interesting to find a suitable extension.

Problem 6. Modify Young’s method and apply it to Schrodinger cocycles.

5. Absence of AC Spectrum for Rough Potentials

It is in some way surprising that a rough sampling function f can make theresulting potentials non-deterministic. Traditionally, non-determinism had beenthought of as a feature induced by the underlying dynamics. In particular, quasi-periodic potentials (in the generalized sense considered in this paper, which allowsdiscontinuous f ’s) had for a long time been considered deterministic. The situationchanged with an important observation by Kotani in his short 1989 paper [46].

He proved the following very general result:

Theorem 9 (Kotani 1989). Suppose that (Ω, T, µ) is ergodic, f : Ω → R takesfinitely many values, and the resulting potentials Vω are µ-almost surely not peri-odic. Then, Leb (Z) = 0 and therefore Σac = ∅.

In particular, operators with quasi-periodic potentials of the form

Vω(n) = λN∑

m=1

γmχ[am−1,am)(nα+ ω),

where 0 = a0 < a1 < · · · aN−1 < aN = 1, γ1, . . . , γN ∈ R (taking at leat two values)and λ 6= 0, have no absolutely continuous spectrum. Note that this result holds forall non-zero couplings and hence it is particularly surprising for small values of λ.We will have more to say about these potentials in the next section.

The family Vωω∈T does not seem to be non-deterministic in an intuitive sense asω is uniquely determined by the sequence Vω|Z− . However, the family becomes non-deterministic when we pass to the closed topological support of the induced measureon λγ1, . . . , λγNZ. Then we will indeed find two distinct sequences that belongto the support of the induced measure, whose restrictions to Z− coincide. This factwill become more transparent when we discuss Theorem 10 below. Kotani’s proofproceeded in a slightly different way; he proved that there can be no continuousmapping from the left half-line to the right half-line.

Proof of Theorem 9. As above we denote the push-forward of µ under the mapω 7→ Vω by ν. It suffices to show that Leb (Z) > 0 implies that supp ν is finite sincethen all elements of supp ν are periodic.

By the continuity of the maps E± from (30) and the fact that Ran f is finite,there is a finite M such that the knowledge of V (−M), . . . , V (−1) determines V (0)uniquely for V ∈ supp ν. Now shift and iterate! It follows that V (−M), . . . , V (−1)completely determine V (n) : n ≥ 0 and hence supp ν has cardinality at most(# Ran f)M .

Consider the case where Ω is a compact metric space, T is a homeomorphism, andµ is an ergodic Borel probability measure. This covers most, if not all, applicationsof interest. In this scenario, Damanik and Killip realized in [26] that finite range off is not essential. What is important, however, is that f is discontinuous at some

18 DAVID DAMANIK

point ω0 ∈ Ω. One can then use this point of discontinuity to actually “construct”two elements of supp ν that coincide on a half-line.

We say that l ∈ R is an essential limit of f at ω0 if there exists a sequence Ωkof sets each of positive measure such that for any sequence ωk with ωk ∈ Ωk,both ωk → ω0 and f(ωk) → l. If f has more than one essential limit at ω0, we saythat f is essentially discontinuous at this point.

Theorem 10 (Damanik-Killip 2005). Suppose Ω is a compact metric space, T :Ω → Ω a homeomorphism, and µ an ergodic Borel probability measure. If thereis an ω0 ∈ Ω such that f is essentially discontinuous at ω0 but continuous at allpoints Tnω0, n < 0, then Σac = ∅.Proof. We again denote the induced measure by ν. For each essential limit l of f atω0, we will find Vl ∈ supp ν with Vl(0) = l. By assumption and construction, Vl(n)is independent of l for every n < 0. This shows Leb (Z) = 0 and hence Σac = ∅.

Let l be an essential limit of f at ω0 and let Ωk be a sequence of sets whichexhibits the fact that l is an essential limit of f . Since each has positive µ-measure,we can find points ωk ∈ Ωk so that Vωk

is in supp ν; indeed, this is the case foralmost every point in Ωk.

As ωk → ω0 and f is continuous at each of the points Tnω0, n < 0, it followsthat Vωk

(n) → Vω0(n) for each n < 0. Moreover, since f(ωk) converges to l, we alsohave Vωk

(0) → l. We can guarantee convergence of Vωk(n) for n > 0 by passing to

a subsequence because RZ is compact. Let us denote this limit potential by Vl. Aseach Vωk

lies in supp ν, so does Vl; moreover, V (0) = l and Vl(n) = Vω0(n) for eachn < 0.

Here is an illustration of this result and a strengthening of the derived conse-quence:

Corollary 5. Suppose Ω = T, µ is normalized Lebesgue measure, and Tω = ω+αfor some irrational α. If f has a single (non-removable) discontinuity at ω0, thenfor all ω ∈ [0, 1), the operator Hω has no absolutely continuous spectrum.

Proof. Let us say that l is a limiting value of f at ω0 if there is a sequence ωk inT\ω0 such that ωk → ω0 and f(ωk) → l. As f has a non-removable discontinuityat ω0, it has more than one limiting value at this point. Moreover, since f iscontinuous away from ω0, any limiting value is also an essential limit since we canchoose each Ωk to be a suitably small interval around ωk.

This shows that f has an essential discontinuity at ω0. As the orbit of ω0

never returns to this point, f is continuous at each point Tnω0, n 6= 0. Therefore,Theorem 10 is applicable and shows that Hω has no absolutely continuous spectrumfor Lebesgue-almost every ω ∈ [0, 1).

It remains to show that the absolutely continuous spectrum of Hω is empty forall ω ∈ R/Z. We begin by fixing ω1 such that Hω1 has no absolutely continuousspectrum and such that the orbit of ω1 does not meet ω0; almost all ω1 have theseproperties.

Given an arbitrary ω ∈ R/Z, we may choose a sequence of integers ni so thatTni(ω) → ω1. As f is continuous on the orbit of ω1, the potentials associated toTni(ω) converge pointwise to Vω1 . By a result of Last and Simon [51], the absolutelycontinuous spectrum cannot shrink under pointwise approximation using translatesof a single potential. Thus, the operator with potential Vω cannot have absolutelycontinuous spectrum. This concludes the proof.


These results are particularly interesting in connection with Problem 2. Quasi-periodic potentials (as defined in this paper) are almost periodic if and only iff is continuous. Thus, proving the absence of absolutely continuous spectrumfor quasi-periodic potentials with discontinuous f ’s is a way of providing furthersupport for the conjecture that Σac 6= ∅ implies almost periodicity.

Let us now turn to the case of continuous sampling functions f . The proofof Theorem 10 certainly breaks down and it is not clear where some sort of non-determinism should come from in the quasi-periodic case, for example. Of course,absence of absolutely continuous spectrum does not hold for a general continuousf . Thus, the following result from [2] is somewhat surprising:

Theorem 11 (Avila-Damanik 2005). Suppose Ω is a compact metric space, T :Ω → Ω a homeomorphism, and µ a non-atomic ergodic Borel probability measure.Then, there is a residual set of functions f in C(Ω) such that Σac(f) = ∅.

Recall that a subset of C(Ω) is called residual if it contains a countable intersec-tion of dense open sets. A residual set is locally uncountable.

One would expect some absolutely continuous spectrum for weak perturbationswith sufficiently nice potentials; especially in the one-frequency quasi-periodic case.More precisely, if f is nice enough, then ∆ + λf(nα+ ω) should have some/purelyabsolutely continuous spectrum for |λ| sufficiently small. It is known that real-analyticity is sufficiently “nice enough” [15] (when α is Diophantine), but it wasexpected that this assumption is much too strong and could possibly be replacedby mere continuity. The proof of Theorem 11 can easily be adapted to yield thefollowing result, also contained in [2], which shows that continuity of the samplingfunction is not sufficient to ensure the existence of absolutely continuous spectrumfor weakly coupled quasi-periodic potentials.

Theorem 12 (Avila-Damanik 2005). Suppose Ω is a compact metric space, T :Ω → Ω a homeomorphism, and µ a non-atomic ergodic Borel probability measure.Then, there is a residual set of functions f in C(Ω) such that Σac(λf) = ∅ foralmost every λ > 0.

Proof. We only sketch the proofs of Theorems 11 and 12. The key technical issueis to establish that the maps

(31) (L1(Ω) ∩Br(L∞(Ω)), ‖ · ‖1) → R, f 7→ Z(f)

and

(32) (L1(Ω) ∩Br(L∞(Ω)), ‖ · ‖1) → R, f 7→∫ Λ

0

Z(λf) dλ

are upper semi-continuous. Here, Λ > 0, Br(L∞(Ω) = f ∈ L∞(Ω) : ‖f‖∞ < r,and Z(f) denotes the set of energies for which the Lyapunov exponent associatedwith (Ω, T, µ, f) vanishes.

Upper semi-continuity of the map (31) can be shown using the fact that γ isharmonic in the upper half-plane and subharmonic on the real line; see [2] fordetails. Fatou’s Lemma then implies upper semi-continuity of (32).

For δ > 0, define

Mδ = f ∈ C(Ω) : Leb(Z(f)) < δ.

20 DAVID DAMANIK

By the upper semi-continuity statement above, Mδ is open. By approximationwith discontinuous functions and upper semi-continuity again, we see that Mδ isalso dense.

It follows that

f ∈ C(Ω) : Σac(f) = ∅ = f ∈ C(Ω) : Leb(Z(f)) = 0 =⋂δ>0

Mδ

is residual and Theorem 11 follows. Given upper semi-continuity of (32), the proofof Theorem 12 is analogous.

While continuous functions can be approximated in the C0 norm by discontinu-ous functions, this does not work in the Cε norm for any ε > 0. Thus, the proof justgiven does not extend to Holder classes. It would be interesting to explore possibleextensions of the results themselves; thus motivating the following problem.

Problem 7. Prove or disprove statements like the ones in Theorems 11 and 12 forHolder classes Cε(Ω), ε > 0.

6. Uniform Lyapunov Exponents and Zero-Measure Spectrum

The Kotani result for potentials taking finitely many values, Theorem 9, is centralto the study of one-dimensional quasi-crystal models. The main results in this areahave been reviewed in [24, 25, 63]. In this section we will therefore focus on therecent progress and discuss why zero-measure spectrum is a consequence of Kotanitheory when there is uniform convergence to the Lyapunov exponent.

One-dimensional quasi-crystals are typically modelled by sequences over a finitealphabet which are aperiodic but which have very strong long-range order prop-erties. An important class of examples is given by one-frequency quasi-periodicpotentials with step functions as sampling functions. That is, the potentials are ofthe form

(33) Vω(n) = λ

N∑m=1

γmχ[am−1,am)(nα+ ω),

where 0 = a0 < a1 < · · · < aN = 1 is a partition of the unit circle, λ, γ1, . . . , γN arereal numbers, α is irrational, and ω ∈ T. We obtain aperiodic potentials if λ 6= 0and γ1, . . . , γN has cardinality at least two. We will assume these conditionsthroughout this section.

One of the properties that has been established for many quasi-crystal modelsis zero-measure spectrum. By general principles, this implies that the spectrum isa Cantor set because it cannot contain isolated points. Given the Kotani result,Leb (Z) = 0, the natural way of proving this is via the identity Σ = Z.

Theorem 13 (Damanik-Lenz 2006). Suppose the potentials are of the form (33)and in addition all discontinuity points am ⊂ T are rational. Then, the Lyapunovexponent vanishes identically on the spectrum, that is, Σ = Z. As a consequence,the spectrum is a Cantor set of zero Lebesgue measure.

Proof. We only sketch the main ideas. More details can be found in [28, 29]. Denote

UH =E : 1

n log ‖AEn (ω)‖ → γ(E) > 0 uniformly in ω

.

Then, by Lenz [53] (see also Johnson [40]), UH = C \ Σ. In particular,

Σ = Z ∪NUH,


whereNUH = E : γ(E) > 0 and E 6∈ UH .

The result follows once NUH = ∅ is established. Thus, given E with γ(E) > 0, weneed to show that 1

n log ‖AEn (ω)‖ → γ(E) uniformly in ω.

Uniform convergence along a special subsequence, nk →∞, can be shown usingresults from Boshernitzan [13] and Lenz [52]. Namely, the assumption that all am

are rational implies that there is a sequence of integers nk →∞ such that for eachk, all words of length nk that occur in the potentials Vω do so with comparablefrequencies That is, there is a uniform C > 0 such that for every k,

(34) min|w|=nk,w occurs

lim infJ→∞

1J

#j : 1 ≤ j ≤ J, Vω(j) . . . Vω(j + nk − 1) = w ≥ C

nk

uniformly in ω [13]. Using this result, one can then use ideas from [52] to show that1

nklog ‖AE

nk(ω)‖ → γ(E) as k →∞, uniformly in ω.

Finally, the avalanche principle of Goldstein and Schlag [35] allows one to in-terpolate and prove the desired uniform convergence of 1

n log ‖AEn (ω)‖ to γ(E) as

n→∞.

The rationality assumption in Theorem 13 holds on a dense set of parameters,which makes it suitable for an approximation of a continuous sampling function bya sequence of step functions. Some consequences that may be drawn from this canbe found in Bjerklov et al. [12]. On the other hand, the assumption is certainlynot necessary and more general results than the one presented here can be foundin [29]. It would be nice if the assumption could be removed altogether:

Problem 8. Prove zero-measure spectrum for all finite partitions of the circle, thatis, remove the rationality assumption from Theorem 13.

It should be mentioned, however, that the proof sketched above will not work inthis generality. The approach is based on the Boshernitzan condition (34), and itwas shown in [29] that this condition fails for certain parameter values.

From a mathematical point of view, the following problem is natural:

Problem 9. Study multi-frequency analogs. That is, for a finite partition Td =J1 ∪ · · · ∪ JN and the operators with potential

Vω(n) = λN∑

m=1

γmχJm(nα+ ω),

what is the measure of the spectrum and what is the spectral type?

These potentials are not directly motivated by quasi-crystal theory but theyform an interesting class that may again prove useful in the understanding of thephenomena that arise for continuous sampling functions. Moreover, very little isunderstood about the associated operators apart from the general Kotani result,which says that there is never any absolutely continuous spectrum. It is unclear,however, whether there can be any point spectrum, for example.

7. Purely AC Spectrum for the Subcritical AMO

In this final section, we describe the (to the best of our knowledge) first appli-cation of Corollary 1. In his 1997 paper, Kotani writes that at the time Kotani

22 DAVID DAMANIK

theory was developed, “it was not clear whether we could know the pure abso-lute continuity ... only from the IDS, but this corollary has answered this questionaffirmatively.”

The application we will present involves the almost Mathieu operator

[Hωψ](n) = ψ(n+ 1) + ψ(n− 1) + 2λ cos(2π(nα+ ω))ψ(n),

that is, the Schrodinger operator with one-frequency quasi-periodic potential asso-ciated with the sampling function f(ω) = 2λ cos(2πω). This operator is known toexhibit a metal-insulator transition at |λ| = 1, that is, for almost every α, the al-most sure spectral type is purely absolutely continuous for |λ| < 1, purely singularcontinuous for |λ| = 1, and pure point (with exponentially decaying eigenfunctions)for |λ| > 1. See Jitomirskaya [38] for this result and its history. For |λ| > 1, one in-deed has to exclude a zero-measure set of frequencies α since it was shown by Avronand Simon, using Gordon’s Lemma, that for Liouville α, there are no eigenvalues[8, 9]. For |λ| ≤ 1, it has long been expected that the results above extend to allirrational frequencies. For |λ| = 1, the issue was resolved by Avila and Krikorian[6]. The question of what happens for |λ| < 1 was addressed by Problem 6 in [60].

Since Bourgain and Jitomirskaya showed in [16] that the Lyapunov exponentassociated with the almost Mathieu operator obeys γ(E) = max0, log |λ| forevery E ∈ Σ for all irrational frequencies α, the problem reduces to a study ofthe integrated density of states, that is, to a proof of its absolute continuity. Thefollowing result was shown in [3] and it completely settles this regularity issue forthe integrated density of states.

Theorem 14 (Avila-Damanik 2006). The integrated density of states of the almostMathieu operator is absolutely continuous if and only if |λ| 6= 1.

Combining this result with the one from [16] just quoted along with Corollary 1,we obtain almost surely purely absolutely continuous spectrum for the subcritical(i.e., |λ| < 1) almost Mathieu operator:

Corollary 6. If |λ| < 1, then Σsing = ∅.

Subsequently, Avila even extended this result and proved purely absolutely con-tinuous spectrum for |λ| < 1, α irrational, and every ω [1].

Problem 10. Extend the results of this section to more general f ∈ L∞(T); forexample, real-analytic f ’s.

It was shown by Bourgain and Jitomirskaya that for Diophantine frequency αand analytic f , Σsing(λf) = ∅ for λ sufficiently small [15]. An extension of thisresult to all ω is contained in [5]. The proofs probably break down for Liouvillefrequencies (for this, one needs to extend the method based on Gordon’s Lemma tomatrices that are not banded, but which do have exponential off-diagonal decay).Thus, it seems natural to attack the problem for Liouville frequencies in the sameway as above.

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Mathematics 253–37, California Institute of Technology, Pasadena, CA 91125, USA

E-mail address: [email protected]

URL: http://www.math.caltech.edu/people/damanik.html

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